The ideas in physics are often much more far-reaching. Their importance often exceeds the impact of philosophical or religious dogmas. In fact, some ideas in physics are so far-reaching that they determine not only the answers but the character of the very questions that are going to be asked for many decades or centuries after the original ideas or principles are revealed. Sometimes we talk about "beauty" in physics but different people often have different features of the theories in mind.
I would prefer to be slightly more specific - as specific as a philosophical essay allows one to be. Some of the main values that may determine the depth of principles and theories in physics may be described as follows:
- uniqueness & rigidity
- self-consistency and mutual causal relations between different statements
- ability to avoid inconsistencies, especially if consistency is not guaranteed from the beginning; equivalently: the existence of miraculous cancellations of inconsistencies
- limited number of independent assumptions
- ability to be relevant in a large number of situations
- power to organize previous systems of ideas and reveal new relations between them
- multiplicity of descriptions of the same structure that are mathematically equivalent
- the maximal possible yet finite amount of complexity of these theories that still makes them compatible with fundamental principles
While I find it obvious that all of these features are important and contribute to the feeling that an idea or a whole framework is on the right track, each of these characteristics has its foes. In fact, there are many people who will tell you that each of these properties is in fact a disadvantage.
Many people will tell you that their proposed theory is good because it is not unique. They will try to convince you that their theory is good exactly because one can deform it in a huge number of ways. One can add anything to these theories. Well, I beg to differ. Valuable theories are always very special animals. Given some assumptions or principles - that must themselves be deep and we will mention some examples later in the text - a valuable theory must be unique or nearly unique. It must look like a golden needle in a haystack. And the criteria that determine which particle is a golden needle must be universal and robust, too.
The mathematical ideas of calculus invented by Newton and Leibniz were inevitable and in some sense unique. The actions defining laws of classical mechanics are not too numerous. The rules of field theory become even more constrained once we assume a beautiful principle such as the Lorentz symmetry. The principles of quantum mechanics are the unique extension of the naive classical deterministic laws that is compatible with the basic rules of logic. The Lagrangians for renormalizable field theories are extremely special because generic Lagrangians lead to a breakdown of the theories at higher energies.
Yang-Mills interactions are inevitable because they are the only known way how the interactions may grow weaker at shorter distances. Their structure requires us to consider theories with local gauge symmetries which are themselves "beautiful" and constrain the matter spectrum and the character of the interactions.
Let me now assume that the reader understands that the theories must be very special and predict a maximal number of consequences from a minimal number of independent assumptions. We don't necessarily need to minimize the number of assumptions and independent concepts as much as we can. Instead, we may work with a larger number of assumptions and objects, but we must always appreciate when some of the assumptions imply others - so that the set of assumptions does not feel disconnected.
It is not "bad circular reasoning" but rather "good rigidity"
Much like all other positive values in physics, connectedness of the ideas has its enemies. Some people will argue that a theory is not good enough because it is based on circular reasoning. This is an argument that some philosophers are ready to use against intellectual structures that are as essential as the theory of relativity - a system of ideas that is more impressive than all ideas ever invented by all philosophers combined.
Is it bad that different statements in relativity follow from each other? Not at all. Every theory must have some assumptions. And if the assumptions tend to imply each other - while still being able to underlie non-trivial and numerous predictions that may even confirm the experiments - we should definitely be more happy, not less happy.
Quantum field theories have the feeling of "relative uniqueness" and string theory offers an "absolute uniqueness" - the first theory the humans ever knew that is capable to do so. This statement does not mean that string theory only has one solution. Of course, it can have many "classical" solutions, asymptotic conditions, and huge Hilbert spaces built upon them. But the dynamical laws that determine which of these spacetimes and states exist and how they evolve are completely unique.
Whenever I talk with someone outside the mainstream high-energy physics, it becomes very clear that many people at the suburbs of the field have absolutely no understanding for these basic principles.
Simplicity and beauty
Another notion that very many people seem to misunderstand is simplicity. Are the laws of Nature simple? Well, it depends how you define the word "simple". We may start with the definition that the laws of Nature are accessible to people with a minimum required IQ, education, or effort. Are the laws of physics simple in this sense? No way. Indeed, as our understanding of Nature deepens, one must - fortunately or unfortunately - learn ever more profound concepts that become increasingly inaccessible to ordinary people and sometimes even to the experts.
Simplicity vs. brevity
Let us try another definition of simplicity. Simplicity may also be defined as the number of T-shirts or pages that we need in order to print all essential rules that will be comprehensible to an intelligent reader. Is Nature simple in this sense? Well, this definition is closer to the truth than the previous one. Indeed, many defining formulae of physics are very short and efficient.
But some of them may be rather long - but equally or more beautiful or fundamental. For example, the full Lagrangian of 11-dimensional supergravity is a rather complicated monster but it does not make it any less beautiful. What is important is that the individual terms in the Lagrangian are not independent from each other. In fact, the whole structure of the two-derivative Lagrangian is uniquely determined by the requirement of supersymmetry.
One of the comments in the list above was a variation of Gell-Mann's totalitarian principle: everything that is not forbidden is compulsory. We should always consider the most general theory that respects the same symmetries and other principles and is equally consistent as the first idea in the class of theories that we have encountered, discovered, or invented. We should never focus on a special, randomly chosen theory that does not differ from many others by the validity of any fundamental principle. We should never try to fool ourselves and others by pretending that such a randomly chosen representative is more important than other random elements of the same set. And we should never study theories with infinitely many parameters, all of which are strongly relevant for the questions that the theory is supposed to answer.
While Gell-Mann would probably think about our duty to add all couplings that respect the rules of the game, it is true that we must also accept all discrete choices for our theories that respect the same principles. Some of these theories may look rather complicated - for example, some people may think that the "E8 x E8" group of a ten-dimensional heterotic string background is too large - but similar theories are important nevertheless because of much more crucial reasons that a naive notion of simplicity.
Symmetries and other valuable principles
When I said "supersymmetry", it is indeed a good opportunity to mention what are the important principles that distinguish which object is the golden needle in a haystack of ideas. Some of the previous "valuable ideas" could be added to this list if we interpret them slightly differently. But we must also include more concrete principles that seem to be absolutely true in the real world, according to everything we know:
- Basic postulates of quantum mechanics - observables are linear operators on a Hilbert space and time evolution is given by another linear operator
- The evolution operators must be unitary to preserve the total probability since the probabilities are computed from squared complex amplitudes
- Dynamics must be local, at least approximately; we discussed locality here; locality is related to causality, another important principle
- Important symmetries should be respected
Symmetries of course play an important role in the scheme of things. Which symmetries do we mean? First of all, it turned out that the discrete symmetries are not terribly constraining and Nature does not care about them too much. People used to think that there was no difference between the left hand and the right hand; between particles and antiparticles. People used to think that C, P, T, and CP were symmetries of Nature.
Today we know better. These symmetries are broken and only the CPT symmetry seems to be the ultimate rule that survived. And it is only true because it can be interpreted as a particular element of a properly extended continuous symmetry, namely the Lorentz symmetry.
The Lorentz symmetry
Well, the Lorentz symmetry and its affine extension, the Poincaré symmetry, is an extremely important principle. This symmetry is a fundamental pillar of special relativity and it includes, via Noether's theorem, the conservation of energy, momentum, angular momentum, and the uniform motion of the center of mass. This symmetry relates many physical phenomena that were considered to be independent: magnetism is an inevitable supplement of electricity once the Lorentz symmetry is assumed. The existence of conserved energy follows from the existence of conserved momentum and vice versa. Moreover, the mass and the energy have to be equivalent and convertible to each other. Many other effects and notions that were independent are unified.
General relativity as a generalization of special relativity
Some people are extremely confused about the nature of special relativity and they will tell you that the discovery of general relativity has revoked the constraints imposed by special relativity. But that's another extremely deep misunderstanding of physics. General relativity is called general relativity because it generalizes special relativity; it does not kill it. One of the fundamental pillars of general relativity is the equivalence principle that states that in locally inertial frames, the laws of special relativity must be satisfied by all local phenomena.
The global Poincaré symmetry of special relativity is extended - or generalized - to the local diffeomorphism symmetry of general relativity. All observers are equally good for formulating the laws, not only the inertial observers. For a simple topology of spacetime, the symmetry of special relativity is a very small subgroup of the symmetry group of general relativity. For other topologies such an embedding can be more subtle but it is important to see that the laws of physics are still constrained by equally strong rules like those in special relativity. The idea that the constraints of special relativity may be forgotten after the papers published in 1915 is a symptom of a breathtaking ignorance.
Some of these people will tell you that the Lorentz symmetry is broken by particular backgrounds or solutions in general relativity and therefore it is no longer important. Well, Lorentz symmetry is spontaneously broken by virtually all configurations you can imagine - both in general relativity as well as special relativity - but in both cases, it is a spontaneous symmetry breaking. What is important is that the underlying laws respect the symmetry: the diffeomorphism and the local Lorentz symmetry of physics. A correct theory based on general relativity must admit a Minkowski (or de Sitter or anti de Sitter, which are equally constraining) solution.
Local vs. global symmetries
The diffeomorphism group is a local symmetry and at the quantum level, all states must be invariant (singlets) under all these local symmetries, much like in the Yang-Mills case. This is why the representation theory of the local, infinite-dimensional gauge groups is irrelevant for physics. On the other hand, states do not have to be invariant under global symmetries such as the Lorentz symmetry, which is why the representation theory of the global symmetries is physically important.
Special relativity is embedded in general relativity in such a way that its global symmetries become generalized examples of diffeomorphisms - that are however "large" in the sense that they are not generated by normalizable modes and therefore are not required to keep the physical states invariant. The generators of time translations and similar geometric operations map the points in the asymptotic regions of the spacetime to other points, and this property is enough to revoke the requirement that the physical states must be invariant under such operations. This is why non-zero (ADM) energy and momentum is possible even in the context of general relativity, even at the quantum level.
Even though the details how the requirement of the Lorentz symmetry - and analogously unitarity - is realized in a given formalism may be subtle, it is absolutely essential to realize that these principles must still hold, and if they are replaced by something else, this "something else" must be at least equally constraining as the original symmetries were in the original theories. Sorry to say but those who say that one can simply forget about unitarity or the Lorentz symmetry altogether are simply idiots.
Cheapness of field redefinitions
Some people are impressed, for scientifically unjustifiable reasons, by random field redefinitions and random functions into which various observables are substituted. Imagine that you are an economist and you invent that instead of the interest rate "X", you should study "Gamma(-1/X)". Because you know the Gamma function, you will say that it is an extremely advanced idea in economics to use "Gamma(-1/x)" instead of "X".
Of course, this example is one of the most pathetic example of a meaningless mathematical masturbation that you can invent. Nevertheless, some people apparently like to play with useless and unjustifiable constructions that are not unsimilar to one from the previous paragraph. Note that if you just redefine some things, you don't gain new physics, new ideas, or new principles. What can happen is that it becomes easier to solve a certain problem and/or invent a new idea or a principle. But one can always translate the insights to the previous variables. At the end, one should be using the variables in which the important features of the physical system are most transparent.
If a system of equations looks "simple" in a random unnatural choice of variables - in which the important principles don't look transparent - then such a "simplicity" is a disadvantage, not an advantage.
Inventing a convoluted system of formulae just because someone likes to obscure things is not tolerable in physics. A random new choice of variables is only justified if it allows one to find some important, unique, universally relevant new solutions, or if it is helpful to illuminate the validity of some key principles - such as unitarity, finiteness, Lorentz symmetry, or some kind of duality. Changes of variables that can't do anything like that have no room in physics. One can never assume that a random change of variables will lead to an interesting new physical idea without having any evidence, and one should never believe the people who are building their perceived "depth" on obscuring things by introducing physically unjustifiable changes of variables.
Dualities and multiplicities of equivalent descriptions
Finally, I want to mention dualities as another feature of theories that are likely to be deep and on the right track. Dualities and multiple descriptions of the same physics certainly did not start with string theory.
Classical physics could have been described by the Lagrangians as well as Hamiltonians. One could have used different coordinates and trivially show that the laws were equivalent. More strikingly, quantum mechanics could have been formulated in the Schrödinger picture, the Heisenberg picture, the Dirac mixed picture, or in terms of the Feynman path integral. The wavefunctions could be represented by functions of coordinates, functions of momenta, or discrete columns of complex numbers that encode the amplitudes of various energy eigenstates. All these approaches turned out to be equivalent although it was not obvious from the beginning.
Is it a bad thing for a theory to have many equivalent languages or formalisms? No way. It was one of the extremely important hints that quantum mechanics was a deep structure. Today, the mathematical facts behind these equivalences look trivial to most of us. But we have new, more impressive equivalences in string theory whose validity seems obviously true but whose "proof as clear as skies" is not yet accessible. In the future, people will most likely find new insights that will make the dualities and equivalences between different descriptions of string/M-theory more transparent. But the very fact that there exists some equivalence that is not immediately obvious suggests that there is something really intriguing to study.
The previous sentence contains the word "immediately". Indeed, if the equivalence of two pictures is immediately obvious, we don't want to count the multiplicity of descriptions as an argument for anything or against anything. The fact that we can use different letters or metric conventions or other conventions (including field redefinitions discussed above) does not mean that we necessarily deal with a deep set of ideas. In fact, it is always possible to formulate any kinds of ideas in different languages. We can only conjecture that we are facing an important idea if the equivalence between the different descriptions that superficially looked inequivalent takes some time and reasoning to be revealed.
Much like in the cases of all other important values in physics, many people are so profoundly confused that they count multiplicity of equivalent - but not manifestly equivalent - descriptions as a disadvantage. On the contrary, these people often prefer narrow-minded formalisms that can only be formulated in one way. They like if their proposed theories can't be connected with other ideas. They like if their colleagues have only mastered one small segment of mathematics. They like if their theories can only be written in the Hamiltonian form but not the path integral form. They don't mind if different approaches that normally lead to the same results give contradictory results in their case.
I beg to differ. Theories should be connected with all formalisms we have used to describe all important classes of phenomena in the past. All methods we have learned in the past should have a correct and sharp generalization that allows us to treat the newly proposed theory. The existence of several non-equivalent approaches is always an advantage. It is a practical advantage when we try to solve or understand the theory: much like other symmetries, dualities can help us to solve particular problems. At the same moment, it is a hint that we are uncovering a huge empire in the world of ideas, not just a fiber in haystack.
We should always remember which features of the ideas and theories that we think about suggest that something important is being uncovered, and we should always point out if someone tries to replace deep and viable principles by shallow, random and unjustifiable dogmas.
And that's the memo.